Question

credit. remember to identify the solution.

1. graph:

$y \geq \frac{2}{3}x + 3$
$y > \frac{-4}{3}x - 3$

Explanation:

Step1: Graph first boundary line

The inequality $y \geq \frac{2}{3}x + 3$ has a solid boundary line (due to $\geq$) with slope $\frac{2}{3}$ and y-intercept $(0,3)$. Plot two points: when $x=0$, $y=3$; when $x=3$, $y=\frac{2}{3}(3)+3=5$. Draw a solid line through $(0,3)$ and $(3,5)$.

Step2: Shade first inequality region

Since $y \geq \frac{2}{3}x + 3$, shade the area above the solid line.

Step3: Graph second boundary line

The inequality $y > \frac{-4}{3}x - 3$ has a dashed boundary line (due to $>$) with slope $\frac{-4}{3}$ and y-intercept $(0,-3)$. Plot two points: when $x=0$, $y=-3$; when $x=3$, $y=\frac{-4}{3}(3)-3=-7$. Draw a dashed line through $(0,-3)$ and $(3,-7)$.

Step4: Shade second inequality region

Since $y > \frac{-4}{3}x - 3$, shade the area above the dashed line.

Step5: Identify overlapping solution

The solution is the overlapping shaded region that satisfies both inequalities.

Answer:

1. Solid line for $y=\frac{2}{3}x+3$ (through $(0,3)$ and $(3,5)$), shade above it.
2. Dashed line for $y=\frac{-4}{3}x-3$ (through $(0,-3)$ and $(3,-7)$), shade above it.
3. The solution is the region where the two shaded areas overlap.